0.08/0.14	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.08/0.14	% Command    : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
0.15/0.36	% Computer   : n010.cluster.edu
0.15/0.36	% Model      : x86_64 x86_64
0.15/0.36	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.15/0.36	% Memory     : 8042.1875MB
0.15/0.36	% OS         : Linux 3.10.0-693.el7.x86_64
0.15/0.36	% CPULimit   : 1200
0.15/0.36	% WCLimit    : 120
0.15/0.36	% DateTime   : Tue Jul 13 16:00:21 EDT 2021
0.15/0.36	% CPUTime    : 
5.28/1.11	% SZS status Theorem
5.28/1.11	
5.28/1.13	% SZS output start Proof
5.28/1.13	Take the following subset of the input axioms:
5.28/1.13	  fof(a, conjecture, leq(multiplication(a, multiplication(a, multiplication(a, a))), star(a))).
5.28/1.13	  fof(additive_associativity, axiom, ![A, B, C]: addition(A, addition(B, C))=addition(addition(A, B), C)).
5.28/1.13	  fof(additive_commutativity, axiom, ![A, B]: addition(A, B)=addition(B, A)).
5.28/1.13	  fof(additive_idempotence, axiom, ![A]: addition(A, A)=A).
5.28/1.13	  fof(left_distributivity, axiom, ![A, B, C]: addition(multiplication(A, C), multiplication(B, C))=multiplication(addition(A, B), C)).
5.28/1.13	  fof(multiplicative_associativity, axiom, ![A, B, C]: multiplication(multiplication(A, B), C)=multiplication(A, multiplication(B, C))).
5.28/1.13	  fof(multiplicative_left_identity, axiom, ![A]: multiplication(one, A)=A).
5.28/1.13	  fof(multiplicative_right_identity, axiom, ![A]: A=multiplication(A, one)).
5.28/1.13	  fof(order, axiom, ![A, B]: (B=addition(A, B) <=> leq(A, B))).
5.28/1.13	  fof(right_distributivity, axiom, ![A, B, C]: addition(multiplication(A, B), multiplication(A, C))=multiplication(A, addition(B, C))).
5.28/1.13	  fof(star_unfold_left, axiom, ![A]: leq(addition(one, multiplication(star(A), A)), star(A))).
5.28/1.13	  fof(star_unfold_right, axiom, ![A]: leq(addition(one, multiplication(A, star(A))), star(A))).
5.28/1.13	
5.28/1.13	Now clausify the problem and encode Horn clauses using encoding 3 of
5.28/1.13	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
5.28/1.13	We repeatedly replace C & s=t => u=v by the two clauses:
5.28/1.13	  fresh(y, y, x1...xn) = u
5.28/1.13	  C => fresh(s, t, x1...xn) = v
5.28/1.13	where fresh is a fresh function symbol and x1..xn are the free
5.28/1.13	variables of u and v.
5.28/1.13	A predicate p(X) is encoded as p(X)=true (this is sound, because the
5.28/1.13	input problem has no model of domain size 1).
5.28/1.13	
5.28/1.13	The encoding turns the above axioms into the following unit equations and goals:
5.28/1.13	
5.28/1.13	Axiom 1 (multiplicative_right_identity): X = multiplication(X, one).
5.28/1.13	Axiom 2 (multiplicative_left_identity): multiplication(one, X) = X.
5.28/1.13	Axiom 3 (additive_idempotence): addition(X, X) = X.
5.28/1.13	Axiom 4 (additive_commutativity): addition(X, Y) = addition(Y, X).
5.28/1.13	Axiom 5 (multiplicative_associativity): multiplication(multiplication(X, Y), Z) = multiplication(X, multiplication(Y, Z)).
5.28/1.13	Axiom 6 (additive_associativity): addition(X, addition(Y, Z)) = addition(addition(X, Y), Z).
5.28/1.13	Axiom 7 (right_distributivity): addition(multiplication(X, Y), multiplication(X, Z)) = multiplication(X, addition(Y, Z)).
5.28/1.13	Axiom 8 (left_distributivity): addition(multiplication(X, Y), multiplication(Z, Y)) = multiplication(addition(X, Z), Y).
5.28/1.13	Axiom 9 (order_1): fresh(X, X, Y, Z) = Z.
5.28/1.13	Axiom 10 (order): fresh3(X, X, Y, Z) = true.
5.28/1.13	Axiom 11 (order_1): fresh(leq(X, Y), true, X, Y) = addition(X, Y).
5.28/1.13	Axiom 12 (order): fresh3(X, addition(Y, X), Y, X) = leq(Y, X).
5.28/1.13	Axiom 13 (star_unfold_right): leq(addition(one, multiplication(X, star(X))), star(X)) = true.
5.28/1.13	Axiom 14 (star_unfold_left): leq(addition(one, multiplication(star(X), X)), star(X)) = true.
5.28/1.13	
5.28/1.13	Lemma 15: addition(X, multiplication(Y, X)) = multiplication(addition(Y, one), X).
5.28/1.13	Proof:
5.28/1.13	  addition(X, multiplication(Y, X))
5.28/1.13	= { by axiom 2 (multiplicative_left_identity) R->L }
5.28/1.13	  addition(multiplication(one, X), multiplication(Y, X))
5.28/1.13	= { by axiom 8 (left_distributivity) }
5.28/1.13	  multiplication(addition(one, Y), X)
5.28/1.13	= { by axiom 4 (additive_commutativity) }
5.28/1.13	  multiplication(addition(Y, one), X)
5.28/1.13	
5.28/1.13	Lemma 16: addition(one, multiplication(addition(X, one), star(X))) = star(X).
5.28/1.13	Proof:
5.28/1.13	  addition(one, multiplication(addition(X, one), star(X)))
5.28/1.13	= { by lemma 15 R->L }
5.28/1.13	  addition(one, addition(star(X), multiplication(X, star(X))))
5.28/1.13	= { by axiom 4 (additive_commutativity) R->L }
5.28/1.13	  addition(one, addition(multiplication(X, star(X)), star(X)))
5.28/1.13	= { by axiom 6 (additive_associativity) }
5.28/1.13	  addition(addition(one, multiplication(X, star(X))), star(X))
5.28/1.13	= { by axiom 11 (order_1) R->L }
5.28/1.13	  fresh(leq(addition(one, multiplication(X, star(X))), star(X)), true, addition(one, multiplication(X, star(X))), star(X))
5.28/1.13	= { by axiom 13 (star_unfold_right) }
5.28/1.13	  fresh(true, true, addition(one, multiplication(X, star(X))), star(X))
5.28/1.13	= { by axiom 9 (order_1) }
5.28/1.13	  star(X)
5.28/1.13	
5.28/1.13	Lemma 17: addition(X, addition(X, Y)) = addition(X, Y).
5.28/1.13	Proof:
5.28/1.13	  addition(X, addition(X, Y))
5.28/1.13	= { by axiom 6 (additive_associativity) }
5.28/1.13	  addition(addition(X, X), Y)
5.28/1.13	= { by axiom 3 (additive_idempotence) }
5.28/1.13	  addition(X, Y)
5.28/1.13	
5.28/1.13	Lemma 18: addition(one, star(X)) = star(X).
5.28/1.13	Proof:
5.28/1.13	  addition(one, star(X))
5.28/1.13	= { by lemma 16 R->L }
5.28/1.13	  addition(one, addition(one, multiplication(addition(X, one), star(X))))
5.28/1.13	= { by lemma 17 }
5.28/1.13	  addition(one, multiplication(addition(X, one), star(X)))
5.28/1.13	= { by lemma 16 }
5.28/1.13	  star(X)
5.28/1.13	
5.28/1.13	Lemma 19: addition(X, multiplication(X, Y)) = multiplication(X, addition(Y, one)).
5.28/1.13	Proof:
5.28/1.13	  addition(X, multiplication(X, Y))
5.28/1.13	= { by axiom 1 (multiplicative_right_identity) }
5.28/1.13	  addition(multiplication(X, one), multiplication(X, Y))
5.28/1.13	= { by axiom 7 (right_distributivity) }
5.28/1.13	  multiplication(X, addition(one, Y))
5.28/1.13	= { by axiom 4 (additive_commutativity) }
5.28/1.13	  multiplication(X, addition(Y, one))
5.28/1.13	
5.28/1.13	Lemma 20: multiplication(star(X), addition(X, one)) = star(X).
5.28/1.13	Proof:
5.28/1.13	  multiplication(star(X), addition(X, one))
5.28/1.13	= { by lemma 19 R->L }
5.28/1.13	  addition(star(X), multiplication(star(X), X))
5.28/1.13	= { by lemma 18 R->L }
5.28/1.13	  addition(addition(one, star(X)), multiplication(star(X), X))
5.28/1.13	= { by axiom 6 (additive_associativity) R->L }
5.28/1.13	  addition(one, addition(star(X), multiplication(star(X), X)))
5.28/1.13	= { by axiom 4 (additive_commutativity) }
5.28/1.13	  addition(one, addition(multiplication(star(X), X), star(X)))
5.28/1.13	= { by axiom 6 (additive_associativity) }
5.28/1.13	  addition(addition(one, multiplication(star(X), X)), star(X))
5.28/1.13	= { by axiom 11 (order_1) R->L }
5.28/1.13	  fresh(leq(addition(one, multiplication(star(X), X)), star(X)), true, addition(one, multiplication(star(X), X)), star(X))
5.28/1.13	= { by axiom 14 (star_unfold_left) }
5.28/1.13	  fresh(true, true, addition(one, multiplication(star(X), X)), star(X))
5.28/1.13	= { by axiom 9 (order_1) }
5.28/1.13	  star(X)
5.28/1.13	
5.28/1.13	Lemma 21: addition(X, star(X)) = star(X).
5.28/1.13	Proof:
5.28/1.13	  addition(X, star(X))
5.28/1.13	= { by lemma 18 R->L }
5.28/1.13	  addition(X, addition(one, star(X)))
5.28/1.13	= { by axiom 6 (additive_associativity) }
5.28/1.13	  addition(addition(X, one), star(X))
5.28/1.13	= { by axiom 11 (order_1) R->L }
5.28/1.13	  fresh(leq(addition(X, one), star(X)), true, addition(X, one), star(X))
5.28/1.13	= { by lemma 20 R->L }
5.28/1.13	  fresh(leq(addition(X, one), multiplication(star(X), addition(X, one))), true, addition(X, one), star(X))
5.28/1.13	= { by lemma 16 R->L }
5.28/1.13	  fresh(leq(addition(X, one), multiplication(addition(one, multiplication(addition(X, one), star(X))), addition(X, one))), true, addition(X, one), star(X))
5.28/1.13	= { by axiom 4 (additive_commutativity) R->L }
5.28/1.13	  fresh(leq(addition(X, one), multiplication(addition(multiplication(addition(X, one), star(X)), one), addition(X, one))), true, addition(X, one), star(X))
5.28/1.13	= { by lemma 15 R->L }
5.28/1.13	  fresh(leq(addition(X, one), addition(addition(X, one), multiplication(multiplication(addition(X, one), star(X)), addition(X, one)))), true, addition(X, one), star(X))
5.28/1.13	= { by axiom 12 (order) R->L }
5.28/1.13	  fresh(fresh3(addition(addition(X, one), multiplication(multiplication(addition(X, one), star(X)), addition(X, one))), addition(addition(X, one), addition(addition(X, one), multiplication(multiplication(addition(X, one), star(X)), addition(X, one)))), addition(X, one), addition(addition(X, one), multiplication(multiplication(addition(X, one), star(X)), addition(X, one)))), true, addition(X, one), star(X))
5.28/1.13	= { by lemma 17 }
5.28/1.13	  fresh(fresh3(addition(addition(X, one), multiplication(multiplication(addition(X, one), star(X)), addition(X, one))), addition(addition(X, one), multiplication(multiplication(addition(X, one), star(X)), addition(X, one))), addition(X, one), addition(addition(X, one), multiplication(multiplication(addition(X, one), star(X)), addition(X, one)))), true, addition(X, one), star(X))
5.28/1.13	= { by axiom 10 (order) }
5.28/1.13	  fresh(true, true, addition(X, one), star(X))
5.28/1.13	= { by axiom 9 (order_1) }
5.28/1.13	  star(X)
5.28/1.13	
5.28/1.13	Lemma 22: addition(star(X), multiplication(Y, addition(X, one))) = multiplication(addition(Y, star(X)), addition(X, one)).
5.28/1.13	Proof:
5.28/1.13	  addition(star(X), multiplication(Y, addition(X, one)))
5.28/1.13	= { by lemma 20 R->L }
5.28/1.13	  addition(multiplication(star(X), addition(X, one)), multiplication(Y, addition(X, one)))
5.28/1.13	= { by axiom 8 (left_distributivity) }
5.28/1.13	  multiplication(addition(star(X), Y), addition(X, one))
5.28/1.13	= { by axiom 4 (additive_commutativity) }
5.28/1.13	  multiplication(addition(Y, star(X)), addition(X, one))
5.28/1.13	
5.28/1.13	Lemma 23: addition(star(X), addition(Y, multiplication(star(X), X))) = addition(Y, star(X)).
5.28/1.13	Proof:
5.28/1.13	  addition(star(X), addition(Y, multiplication(star(X), X)))
5.28/1.13	= { by axiom 4 (additive_commutativity) R->L }
5.28/1.13	  addition(star(X), addition(multiplication(star(X), X), Y))
5.28/1.13	= { by axiom 6 (additive_associativity) }
5.28/1.13	  addition(addition(star(X), multiplication(star(X), X)), Y)
5.28/1.13	= { by lemma 19 }
5.28/1.13	  addition(multiplication(star(X), addition(X, one)), Y)
5.28/1.13	= { by lemma 20 }
5.28/1.13	  addition(star(X), Y)
5.28/1.13	= { by axiom 4 (additive_commutativity) }
5.28/1.13	  addition(Y, star(X))
5.28/1.13	
5.28/1.13	Goal 1 (a): leq(multiplication(a, multiplication(a, multiplication(a, a))), star(a)) = true.
5.28/1.13	Proof:
5.28/1.13	  leq(multiplication(a, multiplication(a, multiplication(a, a))), star(a))
5.28/1.13	= { by axiom 12 (order) R->L }
5.28/1.13	  fresh3(star(a), addition(multiplication(a, multiplication(a, multiplication(a, a))), star(a)), multiplication(a, multiplication(a, multiplication(a, a))), star(a))
5.28/1.13	= { by axiom 4 (additive_commutativity) R->L }
5.28/1.13	  fresh3(star(a), addition(star(a), multiplication(a, multiplication(a, multiplication(a, a)))), multiplication(a, multiplication(a, multiplication(a, a))), star(a))
5.28/1.13	= { by axiom 5 (multiplicative_associativity) R->L }
5.28/1.13	  fresh3(star(a), addition(star(a), multiplication(a, multiplication(multiplication(a, a), a))), multiplication(a, multiplication(a, multiplication(a, a))), star(a))
5.28/1.13	= { by axiom 5 (multiplicative_associativity) R->L }
5.28/1.13	  fresh3(star(a), addition(star(a), multiplication(multiplication(a, multiplication(a, a)), a)), multiplication(a, multiplication(a, multiplication(a, a))), star(a))
5.28/1.13	= { by axiom 4 (additive_commutativity) R->L }
5.28/1.13	  fresh3(star(a), addition(multiplication(multiplication(a, multiplication(a, a)), a), star(a)), multiplication(a, multiplication(a, multiplication(a, a))), star(a))
5.28/1.13	= { by lemma 23 R->L }
5.28/1.13	  fresh3(star(a), addition(star(a), addition(multiplication(multiplication(a, multiplication(a, a)), a), multiplication(star(a), a))), multiplication(a, multiplication(a, multiplication(a, a))), star(a))
5.28/1.13	= { by axiom 8 (left_distributivity) }
5.28/1.13	  fresh3(star(a), addition(star(a), multiplication(addition(multiplication(a, multiplication(a, a)), star(a)), a)), multiplication(a, multiplication(a, multiplication(a, a))), star(a))
5.28/1.13	= { by lemma 23 R->L }
5.28/1.13	  fresh3(star(a), addition(star(a), multiplication(addition(star(a), addition(multiplication(a, multiplication(a, a)), multiplication(star(a), a))), a)), multiplication(a, multiplication(a, multiplication(a, a))), star(a))
5.28/1.13	= { by axiom 5 (multiplicative_associativity) R->L }
5.28/1.13	  fresh3(star(a), addition(star(a), multiplication(addition(star(a), addition(multiplication(multiplication(a, a), a), multiplication(star(a), a))), a)), multiplication(a, multiplication(a, multiplication(a, a))), star(a))
5.28/1.13	= { by axiom 8 (left_distributivity) }
5.28/1.13	  fresh3(star(a), addition(star(a), multiplication(addition(star(a), multiplication(addition(multiplication(a, a), star(a)), a)), a)), multiplication(a, multiplication(a, multiplication(a, a))), star(a))
5.28/1.13	= { by lemma 20 R->L }
5.28/1.13	  fresh3(star(a), addition(star(a), multiplication(addition(star(a), multiplication(addition(multiplication(a, a), multiplication(star(a), addition(a, one))), a)), a)), multiplication(a, multiplication(a, multiplication(a, a))), star(a))
5.28/1.13	= { by lemma 21 R->L }
5.28/1.13	  fresh3(star(a), addition(star(a), multiplication(addition(star(a), multiplication(addition(multiplication(a, a), multiplication(addition(a, star(a)), addition(a, one))), a)), a)), multiplication(a, multiplication(a, multiplication(a, a))), star(a))
5.28/1.13	= { by lemma 22 R->L }
5.28/1.13	  fresh3(star(a), addition(star(a), multiplication(addition(star(a), multiplication(addition(multiplication(a, a), addition(star(a), multiplication(a, addition(a, one)))), a)), a)), multiplication(a, multiplication(a, multiplication(a, a))), star(a))
5.28/1.13	= { by axiom 4 (additive_commutativity) R->L }
5.28/1.13	  fresh3(star(a), addition(star(a), multiplication(addition(star(a), multiplication(addition(addition(star(a), multiplication(a, addition(a, one))), multiplication(a, a)), a)), a)), multiplication(a, multiplication(a, multiplication(a, a))), star(a))
5.28/1.13	= { by axiom 6 (additive_associativity) R->L }
5.28/1.13	  fresh3(star(a), addition(star(a), multiplication(addition(star(a), multiplication(addition(star(a), addition(multiplication(a, addition(a, one)), multiplication(a, a))), a)), a)), multiplication(a, multiplication(a, multiplication(a, a))), star(a))
5.28/1.13	= { by axiom 4 (additive_commutativity) }
5.28/1.13	  fresh3(star(a), addition(star(a), multiplication(addition(star(a), multiplication(addition(star(a), addition(multiplication(a, a), multiplication(a, addition(a, one)))), a)), a)), multiplication(a, multiplication(a, multiplication(a, a))), star(a))
5.28/1.13	= { by axiom 7 (right_distributivity) }
5.28/1.13	  fresh3(star(a), addition(star(a), multiplication(addition(star(a), multiplication(addition(star(a), multiplication(a, addition(a, addition(a, one)))), a)), a)), multiplication(a, multiplication(a, multiplication(a, a))), star(a))
5.28/1.13	= { by lemma 17 }
5.28/1.13	  fresh3(star(a), addition(star(a), multiplication(addition(star(a), multiplication(addition(star(a), multiplication(a, addition(a, one))), a)), a)), multiplication(a, multiplication(a, multiplication(a, a))), star(a))
5.28/1.13	= { by lemma 22 }
5.28/1.13	  fresh3(star(a), addition(star(a), multiplication(addition(star(a), multiplication(multiplication(addition(a, star(a)), addition(a, one)), a)), a)), multiplication(a, multiplication(a, multiplication(a, a))), star(a))
5.28/1.13	= { by lemma 21 }
5.97/1.13	  fresh3(star(a), addition(star(a), multiplication(addition(star(a), multiplication(multiplication(star(a), addition(a, one)), a)), a)), multiplication(a, multiplication(a, multiplication(a, a))), star(a))
5.97/1.13	= { by lemma 20 }
5.97/1.13	  fresh3(star(a), addition(star(a), multiplication(addition(star(a), multiplication(star(a), a)), a)), multiplication(a, multiplication(a, multiplication(a, a))), star(a))
5.97/1.13	= { by lemma 19 }
5.97/1.13	  fresh3(star(a), addition(star(a), multiplication(multiplication(star(a), addition(a, one)), a)), multiplication(a, multiplication(a, multiplication(a, a))), star(a))
5.97/1.13	= { by lemma 20 }
5.97/1.13	  fresh3(star(a), addition(star(a), multiplication(star(a), a)), multiplication(a, multiplication(a, multiplication(a, a))), star(a))
5.97/1.13	= { by lemma 19 }
5.97/1.13	  fresh3(star(a), multiplication(star(a), addition(a, one)), multiplication(a, multiplication(a, multiplication(a, a))), star(a))
5.97/1.13	= { by lemma 20 }
5.97/1.13	  fresh3(star(a), star(a), multiplication(a, multiplication(a, multiplication(a, a))), star(a))
5.97/1.13	= { by axiom 10 (order) }
5.97/1.13	  true
5.97/1.13	% SZS output end Proof
5.97/1.13	
5.97/1.13	RESULT: Theorem (the conjecture is true).
5.97/1.14	EOF